Annals of Operations Research

, Volume 133, Issue 1–4, pp 23–46 | Cite as

The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems

Abstract

The DC programming and its DC algorithm (DCA) address the problem of minimizing a function f=gh (with g,h being lower semicontinuous proper convex functions on Rn) on the whole space. Based on local optimality conditions and DC duality, DCA was successfully applied to a lot of different and various nondifferentiable nonconvex optimization problems to which it quite often gave global solutions and proved to be more robust and more efficient than related standard methods, especially in the large scale setting. The computational efficiency of DCA suggests to us a deeper and more complete study on DC programming, using the special class of DC programs (when either g or h is polyhedral convex) called polyhedral DC programs. The DC duality is investigated in an easier way, which is more convenient to the study of optimality conditions. New practical results on local optimality are presented. We emphasize regularization techniques in DC programming in order to construct suitable equivalent DC programs to nondifferentiable nonconvex optimization problems and new significant questions which have to be answered. A deeper insight into DCA is introduced which really sheds new light on DCA and could partly explain its efficiency. Finally DC models of real world nonconvex optimization are reported.

Keywords

DC programming DC algorithms (DCA) DC duality local optimality conditions global optimality conditions polyhedral DC programming regularization techniques 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. An, Le Thi Hoai. (1994). “Analyse numérique des algorithmes de l’optimisation DC. Approches locale et globale. Codes et simulations numériques en grande dimension. Applications.” Thèse de Doctorat de l’Université de Rouen. Google Scholar
  2. An, Le Thi Hoai. (1997). “Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, algorithmes et applications.” Habilitation à Diriger des Recherches, Université de Rouen. Google Scholar
  3. An, Le Thi Hoai. (2000). “An Efficient Algorithm for Globally Minimizing a Quadratic Function under Convex Quadratic Constraints.” Math. Programming Ser. A 87(3), 401–426. Google Scholar
  4. An, Le Thi Hoai. (2003). “Solving Large Scale Molecular Distance Geometry Problem by a Smoothing Technique via the Gaussian Transform an DC Programming.” J. Global Optim. 27, 375–397. Google Scholar
  5. An, Le Thi Hoai and Pham Dinh Tao. (1997). “Solving a Class of Linearly Constrained Indefinite Quadratic Problems by DC Algorithms.” J. Global Optim. 11, 253–285. Google Scholar
  6. An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (1996). “Numerical Solution for Optimization over the Efficient Set by DC Optimization Algorithm.” Oper. Res. Lett. 19, 117–128. CrossRefGoogle Scholar
  7. An, Le Thi Hoai and Pham Dinh Tao. (1998). “A Branch-and-Bound Method via DC Optimization Algorithm and Ellipsoidal Techniques for Box Constrained Nonconvex Quadratic Programming Problems.” J. Global Optim. 13, 171–206. CrossRefGoogle Scholar
  8. An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (1998). “A Combined DC Optimization – Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems.” J. Combin. Optim. 2(1), 9–28. Google Scholar
  9. An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (1999). “Exact Penalty in DC Programming.” Vietnam J. Math. 27(2), 169–178. Google Scholar
  10. An, Le Thi Hoai and Pham Dinh Tao. (1999). “DCA Revisited and DC Models of Real World Nonconvex Optimization Problems.” Technical Report, LMI, Insa-Rouen. Google Scholar
  11. An, Le Thi Hoai and Pham Dinh Tao. (2000a). “DC Programming Approach for Large-Scale Molecular Optimization via the General Distance Geometry Problem.” In Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches. Nonconvex Optimization and Its Applications, Vol. 40. Kluwer Academic, pp. 301–339. Google Scholar
  12. An, Le Thi Hoai and Pham Dinh Tao. (2000b). “Large Scale Molecular Conformation via the Exact Distance Geometry Problem.” In Optimization. Lecture Notes in Economics and Mathematical Systems, Vol. 481. Heidelberg: Springer, pp. 260–277. Google Scholar
  13. An, Le Thi Hoai and Pham Dinh Tao. (2001a). “A Continuous Approach for Large-Scale Linearly Constrained Quadratic Zero-One Programming.” Optim. 50(1,2), 93–120. Google Scholar
  14. An, Le Thi Hoai and Pham Dinh Tao. (2001b). “DC Optimization Approaches via Markov Models for Restoration of Signals (1-D) and (2-D).” In Advances in Convex Analysis and Global Optimization. Nonconvex Optimization and Its Applications, Vol. 54. Kluwer Academic, pp. 300–317. Google Scholar
  15. An, Le Thi Hoai and Pham Dinh Tao. (2001c). “DC Programming Approach and Solution Algorithm to the Multidimensional Scaling Problem.” In From Local to Global Optimization. Nonconvex Optimization and Its Applications, Vol. 53. Kluwer Academic, pp. 231–276. Google Scholar
  16. An, Le Thi Hoai and Pham Dinh Tao. (2002a). “DC Programming Approach for Multicommodity Network Optimization Problems with Step Increasing Cost Functions.” J. Global Optim. 22(1–4), 204–233. Special Issue, dedicated to Professor R. Horst on the occasion of his 60th birthday. Google Scholar
  17. An, Le Thi Hoai and Pham Dinh Tao. (2002b). “DC Programming. Theory, Algorithms, Applications: The State of the Art.” In First International Workshop on Global Constrained Optimization and Constraint Satisfaction. Valbonne-Sophia Antipolis, France, October 2–4, 2002. Research Report, 28 pages. Laboratory of Modeling, Optimization & Operations Research, Insa-Rouen, France (2002). Google Scholar
  18. An, Le Thi Hoai and Pham Dinh Tao. (2003a). “Large Scale Global Molecular Optimization from Exact Distance Matrices by a DC Optimization Approach.” SIAM J. Optim. 14(1), 77–114. CrossRefGoogle Scholar
  19. An, Le Thi Hoai and Pham Dinh Tao. (2003b). “A New Algorithm for Solving Large Scale Molecular Distance Geometry Problems, Applied Optimization.” In Hight Performance Algorithms and Software for Nonlinear Optimization. Kluwer Academic, pp. 276–296. Google Scholar
  20. An, Le Thi Hoai, Pham Dinh Tao, and Nguyen Van Thoai. (2002). “Combination between Local and Global Methods for Solving an Optimization Problem over the Efficient Set.” European J. Oper. Res. 142, 257–270. Google Scholar
  21. An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (2003a). “Simplicially Constrained DC Optimization over the Efficient Set and Weakly Efficient Sets.” J. Optim. Theory Appl. 117(3), 503–531. Google Scholar
  22. An, Le Thi Hoai, Pham Dinh Tao, and Le Dung Muu. (2003b). “A DC Optimization Approach to Mathematical Programming Problems with Linear Equilibrium Constraints.” Submitted. Google Scholar
  23. An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2002). “Towards Tikhonov Regularization of Nonlinear Ill-Posed Problems: A DC Programming Approach.” C.R. Acad. Sci. Paris Ser. I 335, 1073–1078. Google Scholar
  24. An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2003). “Solving Inverse Problems for an Elliptic Equations by D.C. (Difference of Convex Functions) Programming.” J. Global Optim. 25(4), 407–423. Google Scholar
  25. An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2004a). “On the Ill-Posedness of the Trust Region Subproblem.” J. Inverse Ill-Posed Problems, to appear. Google Scholar
  26. An, Le Thi Hoai, Pham Dinh Tao, and Dinh Nho Hao. (2004b). “DC Programming Approach to Tikhonov Regularization for Nonlinear Ill-Posed Problems.” Submitted. Google Scholar
  27. Conn, A.R., N.I.M. Gould, and P.L. Toint. (2000). Trust Region Methods. MPS–SIAM Series on Optimization. Google Scholar
  28. Hiriat Urruty, J.B. (1989). “Conditions nécessaires et suffisantes d’optimalité globule en optimisation de différences de deux fonctions convexes.” C.R. Acad. Sci. Paris Ser. I 309, 459–462. Google Scholar
  29. Hiriat Urruty, J.B. and C. Lemarechal. (1993). Convex Analysis and Minimization Algorithms. Berlin: Springer. Google Scholar
  30. Horst, R., P.M. Pardalos, and Nguyen Van Thoai. (1995). Introduction to Global Optimization. Kluwer Academic. Google Scholar
  31. Horst, R. and Nguyen Van Thoai. (1999). “DC Programming: Overview.” J. Optim. Theory Appl. 2103(1), 1–43. Google Scholar
  32. Horst, R. and Hoang Tuy. (1996). Global Optimization (Deterministic Approaches), 3rd ed. Springer. Google Scholar
  33. Konno, H., Phan Thien Thach and Hoang Tuy. (1999). Optimization on Low Rank Nonconvex Structures. Kluwer Academic. Google Scholar
  34. Laurent, P.J. (1972). Approximation et optimisation. Paris: Hermann. Google Scholar
  35. Mahey, P. and Pham Dinh Tao. (1993). “Partial Regularization of the Sum of Two Maximal Monotone Operators.” Math. Modell. Numer. Anal. (M 2 AN) 27, 375–395. Google Scholar
  36. Mahey, P. and Pham Dinh Tao. (1995). “Proximal Decomposition of the Graph of Maximal Monotone Operator.” SIAM J. Optim. 5, 454–468. CrossRefGoogle Scholar
  37. More, J.J. and Z. Wu. (1997). “Global Continuation for Distance Geometry Problems.” SIAM J. Optim. 8, 814–836. Google Scholar
  38. More, J.J. and Z. Wu. (1996). “Distance Geometry Optimization for Protein Structures.” Preprint MCS-P628-1296, Argonne National Laboratory, Argonne, IL. Google Scholar
  39. Moreau, J.J. (1965). “Proximité et dualité dans un espace Hilbertien.” Bull. Soc. Math. France 93, 273–299. Google Scholar
  40. Pardalos, P.M. and J.B. Rosen. (1987). Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Sciences, Vol. 268. Springer. Google Scholar
  41. Pschenichny, B.N. (1971). Contrôle optimal et jeux differentiels. Rocquencourt: Cahiers de l’IRIA. Google Scholar
  42. Rockafellar, R.T. (1970). Convex Analysis. Princeton: Princeton University. Google Scholar
  43. Rockafellar, R.T. (1976). “Monotone Operators and the Proximal Point Algorithm.” SIAM J. Control Optim. 14, 877–898. Google Scholar
  44. Tao, Pham Dinh. (1975). “Elements homoduaux relatifs à un couple de normes (φ,ψ). Applications au calcul de S φψ(A).” Technical Report, Grenoble. Google Scholar
  45. Tao, Pham Dinh. (1976). “Calcul du maximum d’une forme quadratique définie positive sur la boule unité de la norme du max.” Technical Report, Grenoble. Google Scholar
  46. Tao, Pham Dinh. (1981). “Contribution à la théorie de normes et ses applications à l’analyse numérique.” Thèse de Doctorat d’Etat Es Science, Université Joseph Fourier, Grenoble. Google Scholar
  47. Tao, Pham Dinh. (1984). “Convergence of Subgradient Method for Computing the Bound Norm of Matrices.” Linear Algebra Appl. 62, 163–182. CrossRefGoogle Scholar
  48. Tao, Pham Dinh. (1985). “Algorithmes de calcul d’une forme quadratique sur la boule unité de la norme maximum.” Numer. Math. 45, 377–440. Google Scholar
  49. Tao, Pham Dinh. (1986). Algorithms for Solving a Class of Non Convex Optimization Problems. Methods of Subgradients. Fermat Days 85. Mathematics for Optimization, Elsevier Science Publishers, B.V. North-Holland. Google Scholar
  50. Tao, Pham Dinh. (1988). “Duality in D.C. (Difference of Convex Functions) Optimization. Subgradient Methods.” In Trends in Mathematical Optimization, International Series of Numerical Mathematics, Vol. 84. Birkhäuser, pp. 277–293. Google Scholar
  51. Tao, Pham Dinh and Le Thi Hoai An. (1994). “Stabilité de la dualité lagrangienne en optimisation DC (différence de deux fonctions convexes).” C.R. Acad. Sci. Paris Sér. I 318, 379–384. Google Scholar
  52. Tao, Pham Dinh and Le Thi Hoai An. (1995). “Lagrangian Stability and Global Optimality in Nonconvex Quadratic Minimization over Euclidean Balls and Spheres.” J. Convex Anal. 2, 263–276. Google Scholar
  53. Tao, Pham Dinh and Le Thi Hoai An. (1998). “DC Optimization Algorithms for Solving the Trust Region Subproblem.” SIAM J. Optim. 8, 476–505. CrossRefGoogle Scholar
  54. Tao, Pham Dinh and Le Thi Hoai An. (1997). “Convex Analysis Approach to DC Programming: Theory, Algorithms and Applications.” Acta Math. Vietnamica 22(1), 287–355. Google Scholar
  55. Thach, Phan Thien. (1993a). “DC Sets, D.C. Functions and Nonlinear Equations.” Math. Programming 58, 415–428. CrossRefGoogle Scholar
  56. Thach, Phan Thien. (1993b). “Global Optimality Criteria and Duality with Zero Gap in Nonconvex Optimization Problems.” SIAM J. Math. Anal. 24, 1537–1556. CrossRefGoogle Scholar
  57. Thach, Phan Thien. (1994). “A Nonconvex Duality with Zero Gap and Applications.” SIAM J. Optim. 4, 44–64. CrossRefGoogle Scholar
  58. Thach, Phan Thien. (1996). “On the Degree and Separability of Nonconvexity and Applications to Optimization Problems.” Math. Programming 77, 23–47. Google Scholar
  59. Toland, J.F. (1978). “Duality in Nonconvex Optimization.” J. Math. Anal. Appl. 66, 399–415. CrossRefGoogle Scholar
  60. Toland, J.F. (1979). “On Subdifferential Calculus and Duality in Nonconvex Optimization.” Bull. Soc. Math. France, Mémoire 60, 177–183. Google Scholar
  61. Tuy, Hoang. (1964). “Concave Programming under Linear Constraints.” Translated Soviet Mathematics 5, 1437–1440. Google Scholar
  62. Tuy, Hoang. (1998). Convex Analysis and Global Optimization. Kluwer Academic. Google Scholar
  63. Tuy, Hoang. (2000). “Monotonic Optimization: Problems and Solution Approaches.” SIAM J. Optim. 2, 464–494. Google Scholar
  64. Wu, Z. (1996). “The Effective Energy Transformation Scheme as a Special Continuation Approach to Global Optimization with Application to Molecular Conformation.” SIAM J. Optim. 6, 748–768. CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Laboratoire de I’Informatique Théorique et Appliquée, UFR MIMUniversité de MetzMetzFrance
  2. 2.Laboratory of Modelling, Optimization and Operations ResearchNational Institute for Applied Sciences-RouenMont Saint Aignan CedexFrance
  3. 3.LMIINSA de RouenMont Saint AignanFrance

Personalised recommendations